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Mirrors > Home > MPE Home > Th. List > nlmdsdir | Structured version Visualization version GIF version |
Description: Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
nlmdsdi.v | ⊢ 𝑉 = (Base‘𝑊) |
nlmdsdi.s | ⊢ · = ( ·𝑠 ‘𝑊) |
nlmdsdi.f | ⊢ 𝐹 = (Scalar‘𝑊) |
nlmdsdi.k | ⊢ 𝐾 = (Base‘𝐹) |
nlmdsdi.d | ⊢ 𝐷 = (dist‘𝑊) |
nlmdsdir.n | ⊢ 𝑁 = (norm‘𝑊) |
nlmdsdir.e | ⊢ 𝐸 = (dist‘𝐹) |
Ref | Expression |
---|---|
nlmdsdir | ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋𝐸𝑌) · (𝑁‘𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ NrmMod) | |
2 | nlmdsdi.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | nlmngp2 23216 | . . . . . . 7 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
4 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝐹 ∈ NrmGrp) |
5 | ngpgrp 23135 | . . . . . 6 ⊢ (𝐹 ∈ NrmGrp → 𝐹 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝐹 ∈ Grp) |
7 | simpr1 1186 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ 𝐾) | |
8 | simpr2 1187 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑌 ∈ 𝐾) | |
9 | nlmdsdi.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
10 | eqid 2818 | . . . . . 6 ⊢ (-g‘𝐹) = (-g‘𝐹) | |
11 | 9, 10 | grpsubcl 18117 | . . . . 5 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋(-g‘𝐹)𝑌) ∈ 𝐾) |
12 | 6, 7, 8, 11 | syl3anc 1363 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑋(-g‘𝐹)𝑌) ∈ 𝐾) |
13 | simpr3 1188 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉) | |
14 | nlmdsdi.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
15 | nlmdsdir.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑊) | |
16 | nlmdsdi.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
17 | eqid 2818 | . . . . 5 ⊢ (norm‘𝐹) = (norm‘𝐹) | |
18 | 14, 15, 16, 2, 9, 17 | nmvs 23212 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋(-g‘𝐹)𝑌) ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → (𝑁‘((𝑋(-g‘𝐹)𝑌) · 𝑍)) = (((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌)) · (𝑁‘𝑍))) |
19 | 1, 12, 13, 18 | syl3anc 1363 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑁‘((𝑋(-g‘𝐹)𝑌) · 𝑍)) = (((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌)) · (𝑁‘𝑍))) |
20 | eqid 2818 | . . . . 5 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
21 | nlmlmod 23214 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ LMod) |
23 | 14, 16, 2, 9, 20, 10, 22, 7, 8, 13 | lmodsubdir 19621 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋(-g‘𝐹)𝑌) · 𝑍) = ((𝑋 · 𝑍)(-g‘𝑊)(𝑌 · 𝑍))) |
24 | 23 | fveq2d 6667 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑁‘((𝑋(-g‘𝐹)𝑌) · 𝑍)) = (𝑁‘((𝑋 · 𝑍)(-g‘𝑊)(𝑌 · 𝑍)))) |
25 | 19, 24 | eqtr3d 2855 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌)) · (𝑁‘𝑍)) = (𝑁‘((𝑋 · 𝑍)(-g‘𝑊)(𝑌 · 𝑍)))) |
26 | nlmdsdir.e | . . . . 5 ⊢ 𝐸 = (dist‘𝐹) | |
27 | 17, 9, 10, 26 | ngpds 23140 | . . . 4 ⊢ ((𝐹 ∈ NrmGrp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋𝐸𝑌) = ((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌))) |
28 | 4, 7, 8, 27 | syl3anc 1363 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑋𝐸𝑌) = ((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌))) |
29 | 28 | oveq1d 7160 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋𝐸𝑌) · (𝑁‘𝑍)) = (((norm‘𝐹)‘(𝑋(-g‘𝐹)𝑌)) · (𝑁‘𝑍))) |
30 | nlmngp 23213 | . . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
31 | 30 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ NrmGrp) |
32 | 14, 2, 16, 9 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → (𝑋 · 𝑍) ∈ 𝑉) |
33 | 22, 7, 13, 32 | syl3anc 1363 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑋 · 𝑍) ∈ 𝑉) |
34 | 14, 2, 16, 9 | lmodvscl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → (𝑌 · 𝑍) ∈ 𝑉) |
35 | 22, 8, 13, 34 | syl3anc 1363 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → (𝑌 · 𝑍) ∈ 𝑉) |
36 | nlmdsdi.d | . . . 4 ⊢ 𝐷 = (dist‘𝑊) | |
37 | 15, 14, 20, 36 | ngpds 23140 | . . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝑋 · 𝑍) ∈ 𝑉 ∧ (𝑌 · 𝑍) ∈ 𝑉) → ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)) = (𝑁‘((𝑋 · 𝑍)(-g‘𝑊)(𝑌 · 𝑍)))) |
38 | 31, 33, 35, 37 | syl3anc 1363 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)) = (𝑁‘((𝑋 · 𝑍)(-g‘𝑊)(𝑌 · 𝑍)))) |
39 | 25, 29, 38 | 3eqtr4d 2863 | 1 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋𝐸𝑌) · (𝑁‘𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 · cmul 10530 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 distcds 16562 Grpcgrp 18041 -gcsg 18043 LModclmod 19563 normcnm 23113 NrmGrpcngp 23114 NrmModcnlm 23117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 |
This theorem is referenced by: nlmvscnlem2 23221 |
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